Corollary 2 
For sparse models, when \((\log p)/n\rightarrow 0\) , \(\sigma ^2(s\log p)/(n\Vert \theta _0\Vert ^2)\rightarrow 0\) , \(\widehat{\theta }_0\)  is the LASSO estimate and \(\widehat{\Sigma }\)  is the sparse covariance estimator in [1]}, it satisfies that
\(&&\mathbb {E}\Vert \widehat{\theta }-\theta ^*\Vert ^2=O\left(\frac{s\sigma ^2\log p}{n}+ v^2n^{-\frac{2\alpha }{2\alpha +1}} \right).\) 
